PREDICTION OF LEAKAGE FLOW AND ROTORDYNAMIC ...jestec.taylors.edu.my/Vol 12 issue 11 November...

Journal of Engineering Science and Technology Vol. 12, No. 11 (2017) 2922 - 2938 © School of Engineering, Taylor’s University

2922

PREDICTION OF LEAKAGE FLOW AND ROTORDYNAMIC CHARACTERISTICS FOR AN ECCENTRIC

LABYRINTH GAS SEAL

MOHAMED KAMOUNI

Sidi Mohamed ben Abdellah University, High School of Technology,

Post box 2427, Road of Imouzzer, Fez, Morocco

E-mail: [email protected]

Abstract

Labyrinth seals are key elements to limit leakage flow between rotating and

stationary parts of turbo machines. However, these seals can modify the

rotordynamic stability of machines. Thus, accurate predictions of static and

dynamic behaviour for labyrinth seals are very important to optimize efficiency

and operating conditions of rotating machines using this kind of seals. The

present work contributes by a numerical model based on CFD computation to

predict leakage flow and rotordynamic coefficients for a short eccentric

labyrinth seal with four teeth fixed on the rotor. The developed model accuracy

has been validated on experimental measurements of the pressure distribution

along and around the seal which drops from 110770 Pa at the seal inlet to

103300 Pa at the seal outlet. A parametric study has been conducted to show

the effect of pressure ratio and inlet swirl ratio on leakage flow and

rotordynamic coefficients of the seal. In this study, the outlet pressure is kept

constant but the inlet/outlet pressure ratio varies from 1.072 to 8 while three

inlet swirl ratios (0, 0.5 and 1) are considered. Obtained results of this work are

presented to help designers and industrials optimizing operating conditions and

improving performances of this kind of seals.

Keywords: Labyrinth seal, leakage flow, Rotordynamic coefficients, inlet swirl

velocity, whirl frequency.

1. Introduction

Labyrinth seals are widely used to reduce leakage in various kinds of

turbomachines including turbines, turbo-pumps and compressors. Like bearings in

a rotor assembly, these seals have similar rotordynamic force coefficients and

Prediction of Leakage Flow and Rotordynamic Characteristics . . . 2923

Journal of Engineering Science and Technology November 2017, Vol. 12(11)

Nomenclatures

C Sealing mean clearance, m

D Direct damping cefficient, N.s.m-1

d Cross-coupled damping cefficient, N·s.m-1

e Eccentricity, m

Fr Force component in radial direction, N

Ft Force component in tangential direction, N

Fy Force component in y-direction, N

Fz Force component in z-direction, N

K Direct stiffness coefficient, N.m-1

k Cross-coupled stiffness coefficient, N.m-1

Pin Inlet pressure, Pa

Pout Outlet pressure, Pa

PR Pressure ratio (Pin/Pout)

R Rotor radius, m

t Time, s

ui , uj Velocity components in the coordinate system, m.s-1

Win Inlet preswirl velocity, m.s-1

xi, xj Coordinate system components, m

y, z Position components of the rotor center, m

y , z Temporal derivation of y and z components, m.s-1

Greek Symbols

δ Whirling ratio (Ω/ω)

ε Eccentriciy ratio (e/C)

θ Anglular position of the rotor center relative to the x axis, deg

λ Inlet preswirl ratio (Win/Rω)

μ Dynamic viscosity, kg.m-1

s-1

ρ Density, kg.m-3

Ω Angular frequency of rotor whirling, rad.s-1

ω Angular frequency of rotor spin, rad.s-1

generate fluid response forces that can either improve (stabilizing forces) or

degrade (destabilizing forces) the dynamic stability of rotor systems [1]. These

seal forces, caused by non-uniform circumferential pressure distribution, must be

controlled to ensure that the rotor-bearing system remains stable throughout its

operational envelope. In order to meet the sealing requirement as defined by the

design constraints and required sealing effectiveness, designers have developed

many labyrinth seal types: straight, stepped, staggered, inclined, etc. [2]. The

complex working flow passing through labyrinth seals has been the subject of

many worldwide theoretical and experimental studies in the last three decades.

However, various disagreements have been reported between rotordynamic

predictions and measurements for these seals. Thus, prediction of leakage and

rotordynamic coefficients remains a demanding and challenging task for labyrinth

gas seals [3].

In the field of seals’ rotordynamic experimental research, Benckert and

Wachter [4] were among the first authors to present an early experimental

investigation on the rotordynamic characteristics of straight labyrinth seals using

2924 M. Kamouni


a test rig applying hydraulic shakers to excite the seal stator. They measured the

frequency-independent direct and cross-coupled stiffness coefficients for

different labyrinth geometries and first discussed the effects of pressure ratio,

rotational speed, and inlet flow conditions on the rotordynamic stability of

labyrinth seals. Their investigations have proven that the lateral force component

acting at a right angle to the rotor deflection plane, represents a vibration exciting

force, which has to be accounted for in rotordynamics. They reported that with

increasing shaft circumferential speed - pressure ratio constant -, the lateral force

is growing. Additionally, for short seals, some swirl webs in front of the first

cavity of the seal are sufficient to reduce the preswirl and therefore the lateral

force sensitivity.

Childs and Scharrer [5, 6] and Picardo and Childs [7] measured the

rotordynamic force coefficients of straight labyrinth seals with teeth on rotor or

teeth on stator at various pressure ratios, rotational speeds, sealing clearances and

inlet swirl velocities. Their results show that the teeth on stator labyrinth seal is

more stable due to lower cross-coupled stiffness values. Also, the effective

damping of the labyrinth seal is almost completely insensitive to changes in

sealing clearance. Furthermore, it has been showed that labyrinth seals possess

frequency-independent rotordynamic coefficients up to a frequency 150 Hz.

Kwanka et al. [8] and Pugachev and Deckner [9] measured the stiffness and

damping coefficients of staggered labyrinth seals using a test rig applying

magnetic bearing to excite the seal rotor. They presented the effects of pressure

ratio, inlet swirl, rotational speed, and rotor eccentricity on the seal stiffness

and damping. Their obtained results show that direct stiffness is negative and

static direct stiffness increases strongly with the leakage while the dynamic

direct stiffness varies only slightly. The measured cross-coupled stiffness

decreases with increasing rotational speed while the measured direct stiffness

remains relatively constant. They added that damping coefficients are more

sensitive to the rotor eccentricity values with a maximum deviation of about

20% at the largest eccentricity. To explain some discrepancies between

measurements and predictions for dynamic coefficients, they reported that

direct stiffness is particularly sensitive to the measuring method and the inflow

cavity used for creating the inlet swirl can generate a significant force at

certain operating conditions.

Theoretical rotordynamic modelling of labyrinth seals was mainly based on

the bulk-flow approach which continues to be used in the industry [10]. Iwatsubo

was the first to develop this approach in 1980 [11]. He presented a one-control-

volume bulk-flow model to predict the cavity pressure, circumferential-velocity

distribution, and rotordynamic coefficients of a labyrinth seal. He showed that a

continuous vortex in the circumferential direction occurs in the fluid flow and its

form is like a sinusoidal wave which is rotating in the same direction as the rotor.

The primary advantage of bulk-flow models is that they can predict the seal

rotordynamic coefficients with efficient computational time. However, these

models lack flow details and rely on empirical corrections such as flow

coefficients and friction factors [12, 13] that may change for varying operational

conditions and seal geometry.

Due to the limitations of this approach, the computational fluid dynamic

(CFD) methods are more and more applied everywhere for solving Navier-



Stokes equations to obtain more satisfactory predictions at different boundary

and operating conditions. In a general way, with the increased availability of

computational resources, comprehensive CFD models are now gradually

replacing analytical approaches to predict and simulate thermal and dynamical

behaviours in porous mediums as in the works of Adegun et al. [14] where a

dynamic analysis has been developed for an incompressible flow through a

porous landfill and Pakdee et al. [15] where a premixed combustion has been

simulated in a porous media. Moreover, in the same field, Chand el al. [16] have

also developed a heat convection parametric study in a nanofluid layer in a

porous medium based on Galerkin method.

Currently, several recent works applied commercial CFD codes to labyrinth

gas seal rotordynamic problems [17, 18]. In 2012, Pugachev et al. [18] have used

a full 3D eccentric CFD model built in ANSYS CFX to predict rotordynamic

coefficients of a short staggered three-toot-on-stator labyrinth seal. They reported

that the SST turbulence model shows robust performance, however, for highly

pre-swirled flows, the turbulence production terms should be limited. Otherwise

the SST model over-predicts the cross-coupled stiffness coefficients. Then more

recently, in 2016, Sun et al. [19] have studied the effect of swirl brakes on

labyrinth seals using a CFD model. Their obtained results show that the swirl

brakes play an important role in decreasing the inlet swirl ratio of the seal (60 to

75% less). In addition, by using swirl brakes at the seal entrance, the seal

destabilizing forces (cross-coupled stiffness coefficients) decrease (50 to 300%

less) while the seal direct damping coefficients increase (50 to 60% larger).

From these numerous applications, one can conclude that CFD technique is a

powerful tool, but the accuracy of predictions compared to measurements is not

always sufficient and thus, further research is still needed to make CFD

calculations a standard of the design process for the prediction of rotordynamic

coefficients of labyrinth gas seals. In this context, the present research attempts

to calculate the leakage flow and rotordynamic forces through an eccentric short

labyrinth gas seal based on three-dimensional CFD techniques solving the

general Reynolds Averaged Navier-Stokes equations along with appropriate

turbulence model.

2. Geometry and CFD Model of the Seal

2.1. Seal geometry

The straight labyrinth seal object of this study has four teeth fixed on the rotor

lateral surface. These teeth generate three cavities in the seal. Figure 1 shows the

2-D seal geometry for a centred position of the rotor with respect to the stator.

2926 M. Kamouni


Fig. 1. Labyrinth seal geometry.

The geometrical parameters shown in the 2-D seal geometry are defined and

summarized in Table 1.

Table 1. Dimensions of the seal.

Parameter Value

Rotor radius, R (mm) 93.66

Tooth width, L (mm) 12.7

Tooth thickness, T (mm) 3.18

Tooth height, H (mm) 7.94

Mean clearance width, C (mm) 0.949

The seal working fluid is air. A cut section of the 3-D fluid computational

domain is shown in Fig. 2. The teeth are represented by circumferential grooves

in this computational domain.

Fig. 2. A cut section of the 3-D computational domain of the seal.



2.2. Governing equations

The turbulent flow through the seal is governed by the continuity and momentum

equations.

Continuity equation:

𝜕𝜌

𝜕t+

𝜕(𝜌𝑈𝑖)

𝜕𝑥𝑖= 0 (1)

Momentum equation:

𝜕𝜌𝑈𝑖

𝜕𝑡+

𝜕(𝜌𝑈𝑖𝑈𝑗)

𝜕𝑥𝑗= −

𝜕 𝑝′

𝜕𝑥𝑖+

𝜕

𝜕𝑥𝑗[𝜇𝑒𝑓𝑓 (

𝜕𝑈𝑖

𝜕𝑥𝑗+

𝜕𝑈𝑗

𝜕𝑥𝑖)] + 𝑆𝑀 (2)

where i and j are the indexes relating to the three directions of space, SM is the

sum of body forces and μeff is the effective viscosity accounting for turbulence. p'

is the modified pressure.

𝑝′ = 𝑝 +2

3𝜌𝑘 +

2

3𝜇𝑒𝑓𝑓

𝜕𝑈𝑘

𝜕𝑥𝑘 (3)

𝜇𝑒𝑓𝑓 = 𝜇 + 𝜇𝑡 (4)

where μt is the turbulence viscosity and μ is a constant.

The turbulence model assumes that the turbulence viscosity is linked to the

turbulence kinetic energy and dissipation via the relation:

𝜇𝑡 = 𝐶𝜇𝜌𝑘2

𝜀 (5)

where Cμ is a constant.

The values of k and ε are obtained directly from the differential transport

equations for the turbulence kinetic energy and turbulence dissipation rate:

𝜕(𝜌𝑘)

𝜕𝑡+

𝜕(𝜌𝑈𝑗𝑘)

𝜕𝑥𝑗=

𝜕

𝜕𝑥𝑗[(𝜇 +

𝜇𝑡

𝜎𝑘)

𝜕𝑘

𝜕𝑥𝑗] + 𝑃𝑘 − 𝜌𝜀 + 𝑃𝑘𝑏 (6)

𝜕(𝜌𝜀)

𝜕𝑡+

𝜕(𝜌𝑈𝑗𝜀)

𝜕𝑥𝑗=

𝜕

𝜕𝑥𝑗[(𝜇 +

𝜇𝑡

𝜎𝜀)

𝜕𝜀

𝜕𝑥𝑗] +

𝜀

𝑘(𝐶𝜀1𝑃𝑘 − 𝐶𝜀2𝜌𝜀 + 𝐶𝜀1𝑃𝜀𝑏) (7)

where Cε1, Cε2, σk, and σε are constants. Pkb and Pεb represent the influence of the

buoyancy forces. Pk is the turbulence production due to viscous forces, which is

modelled with:

𝑃𝑘 = 𝜇𝑡 (𝜕𝑈𝑖

𝜕𝑥𝑗+

𝜕𝑈𝑗

𝜕𝑥𝑖)

𝜕𝑈𝑖

𝜕𝑥𝑗−

2

3

𝜕𝑈𝑘

𝜕𝑥𝑘(3𝜇𝑡

𝜕𝑈𝑘

𝜕𝑥𝑘+ 𝜌𝑘) (8)

2.3. Boundary conditions

The boundary conditions of the seal are as it follows:

2928 M. Kamouni


The rotor surface has a circumferential velocity which is the product of rotor

rotational speed and rotor radius. The stator surface is static. Standard wall

boundary conditions are used for fluid-solid interaction regions.

At the seal entrance, the inlet pressure Pin is equal to the upstream reservoir

total pressure and the inlet circumferential velocity win is given

independently (win= 0%, 50% or 100% of Rω).

The flow is adiabatic in the seal.

At the seal outlet, the exit pressure Pout is equal to the downstream reservoir

static pressure.

2.4. Meshing

For the given seal geometry, an appropriate mesh is required to describe correctly

the flow within the seal. Hexahedral mesh elements were used to create three

dimensional non-uniform structured meshes in the entire computational domain.

An adequate mesh refinement is allowed to the clearance area and boundary

layers to accurately calculate desired physical parameters of the flow within the

seal. Figure 3 shows a cut section of generated computational grids in the 3-D

computational domain.

Fig. 3. A cut section of the 3-D Mesh used

for the fluid computational domain of the seal.

2.5. Rotating frame

Observing the motion of rotor-seal system from a stationary frame, the rotor is

spinning at the speed ω while also whirling at the speed Ω at the same time,

which means that the location of rotor and thus the shape of mesh are changing

all the time. So, it is actually a transient problem involved with mesh moving. To

avoid a transient analysis and moving mesh, a rotating frame with the speed Ω

was applied as shown in Fig. 4. In the rotating frame, the rotor itself spins at the



speed (ω-Ω), while the stator spins at the speed Ω in the opposite direction to the

frame. Thus, it becomes a steady state problem and there is no mesh moving.

Fig. 4. Frame transfer from stationary to rotating.

2.6. Rotordynamic coefficients

The fluid driving forces acting on the rotor of the seal are represented in Fig. 5.

These forces can be obtained at each whirl frequency Ω by integration of the

pressure along and around the seal rotor surface.

dx dθ sθco pRFy (9)

dx dθ θ sinpRFz (10)

Fig. 5. Fluid forces induced in the seal.

2930 M. Kamouni


A positive radial force is a centring force while a negative radial force is a

decentring one. A positive tangential force is a forward whirl force while a

negative tangential force is a backward whirl force. Rotordynamic instability

occurs when the forward driving forces exceed the resisting dissipation forces,

which leads to self-excitation of the first whirling mode of the rotor [19].

For small motion about a centred position, the relation between the reaction-

force components and the shaft motion is defined by the following linearized

dynamic model.

z

y

Dd

dD

z

y

Kk

kK

F

F-

z

y (11)

where (y, z) define the lateral motion of the rotor relative to the stator, (FY , FZ) are

the components of the reaction force acting on the rotor. (K, k) and (D, d)

represent the direct and cross-coupled stiffness and damping coefficients,

respectively. The cross-coupled terms (k, d) arise from the fluid’s circumferential

velocity component. If the shaft centre moves in a circular orbit with rayon e, then

the rotation displacement vector of the shaft centre has coordinates:

t)(Ωcos.ey (12)

t)(Ωsin.ez (13)

The force components acting on the rotor (Fig. 5) can be given from the

dynamic equation (11), when t=0, as following:

K-Ω d -=0)=(tF)(F

ee

yr

(14)

DΩ-k=e

0)=(tF

e

)(F zt

(15)

The direct stiffness coefficient K and the cross coupled damping coefficient d

can be obtained employing a linear regression of the calculated radial force for two

values of Ω (e.g., Ω = 0 and Ω = 0.5ω) in Eq. (14). The direct damping coefficient

D and the cross coupled stiffness coefficient k can be determined from linear

regression of the calculated tangential force for two values of Ω (e.g., Ω = 0 and Ω

= 0.5ω) in Eq. (15).

3. Results and Discussions

3.1. Validation

To ensure the validity and accuracy of the calculations, the results are compared to

available experimental results made on the same seal by Rajakumar and Sisto [20].

The used operating conditions of the seal are summarized in Table 2. The negative

signs of the rotating speed and the inlet swirl velocity indicate that the rotor turns in

the clockwise direction as per the angle sign convention shown in Fig. 5.

Rajakumar and Sisto [20] have locally measured pressure in each mid-cavity

of the seal. Figures 6 and 7 show a comparison of these measurements and the



current CFD predictions for the static pressure distribution along and around the

seal, respectively. One can easily see that predictions are in good agreement with

experiments. Additionally, Fig. 6 shows that the pressure drops from the inlet

pressure at left to the outlet pressure at right, and the pressure is almost equal in

each cavity interior.

Table 2. Operating conditions of the seal.

Parameter Value

Eccentricity ratio, ε 43 %

Rotating speed, ω (rpm) -2025

Inlet pressure, Pin (Pa) 11077 0

Outlet pressure, Pout (Pa) 103300

Inlet swirl velocity, Win (ms-1

) -49.8

Fig. 6. CFD and experimental pressure

distribution in the axial direction of the seal.

Fig. 7. CFD and experimental pressure distribution

in the circumferential direction at the mid-cavities of the seal.

0,00 0,01 0,02 0,03 0,04 0,05102000

104000

106000

108000

110000

Pre

ssu

re (P

a)

X (m)

P (Pa) CFD

P (Pa) Exp

60 120 180 240 300

104000

105000

106000

107000

108000

109000

Pe

ss

ure

(P

as

ca

l)

cav1 Exp

cav2 Exp

cav3 Exp

cav1 CFD

cav2 CFD

cav3 CFD

2932 M. Kamouni


Figure 8 shows contours of the static pressure in the XY plane of the seal. The

static pressure drop starts from the seal inlet pressure to reach the outlet pressure

at the seal exit. The quasi same colour in each cavity confirms that static pressure

is almost constant in each cavity interior. Although no pressure change in the

tooth cavity, there is a great pressure drop when the stream flows the tooth gap

because the part of the flow is changed into flow speed from static pressure.

Furthermore, it is shown that pressure drop mainly occurs in the left zone of each

cavity at the labyrinth tooth throttling.

Fig. 8. Pressure contours in the XY plane of the seal.

Figure 9 shows the velocity vectors in a radial axial plane of the seal. It can be

easily seen that there is a high speed jet at the tip of each tooth and a large vortex

inside each chamber. The high pressure drop occurs in the first cavity where a strong

flow jet is generated making the flow more turbulent in this cavity. We note that the

recirculation zones in the seal cavities act as brakes to stop the axial velocity of the

flow through the seal and therefore to reduce the leakage flow. These results confirm

the velocity distribution results obtained by Sun et al. [19] in their experimental and

numerical studies of labyrinth seals with inlet swirl brakes.

Fig. 9. Velocity vectors in the XY plane of the seal.



3.2. Effects of pressure ratio, seal clearance and inlet swirl ratio on

leakage flow

Based on the previous operating conditions used to validate the model, we

investigated the static behavior of the labyrinth seal through its leakage rate.

Figure 10 represents leakage versus the pressure ratio (Pin/Pout) with the inlet

swirl ratio as a parameter. This figure shows that the amount of leakage is more

influenced by the inlet/outlet pressure ratio, namely, the leakage increases with

increasing pressure ratio. Additionally, it is shown that leakage flow decreases

very slightly with increasing inlet swirl to the point that we can consider that the

leakage flow through this kind of seals is practically not influenced by the inlet

swirl ratio for the envisaged range of pressure ratio.

Figure 11 represents leakage flow versus the pressure ratio (Pin/Pout) with the

mean radial clearance as a parameter. This figure shows that the leakage

increases with increasing pressure ratio and mean radial clearance. Furthermore,

it can be stated that roughly for this prediction range, the leakage flow increases

nearly in the same proportion with increasing mean radial clearance. In addition,

the leakage flow becomes more sensitive to pressure ratio variations with high

radial clearances. These leakage flow results agree well with the more recent

results obtained by Dan Sun et al. [19] and Zhigang et al. [21] for other labyrinth

seal configurations and operating conditions.

Fig. 10. Leakage flow versus pressure

ratio with the inlet swirl ratio as a parameter.

Fig. 11. Leakage flow versus pressure ratio

with the mean radial clearance as a parameter.

1 2 3 4 5 6 7 80,00

0,01

0,02

0,03

0,04

0,05

Leakag

e (

Kg

/S)

PR = Pin/P

out

= 0

= 0,5

= 1

1 2 3 4 5 6 7 80,00

0,01

0,02

0,03

0,04

0,05

Le

ak

ag

e (

Kg

/S)

PR = Pin/P

out

C = 0,949 mm

C = 0,500 mm

C = 0,250 mm

2934 M. Kamouni


3.3. Effects of pressure and inlet swirl ratios on dynamic coefficients

The following section deals with the dynamic behaviour of the seal. A parametric

study will be conducted with an eccentricity and whirling ratios of 10% and 50%

respectively, to analyse the influence of some operating conditions (pressure and

inlet swirl ratios) on rotordynamic coefficients of the seal. All other operating

conditions remain unchanged.

Figures 12 and 13 represent respectively direct and cross coupled stiffness

coefficients versus the pressure ratio (Pin/Pout) with the inlet swirl ratio as a

parameter. It should be noted that the direct stiffness coefficient is always

negative and both magnitudes of direct and cross couples stiffness coefficients

increase almost linearly with increasing pressure ratio and inlet swirl ratio. These

results confirm those obtained by Pugachev et al. [18] in their numerical study

relating to CFD prediction of rotordynamic coefficients for short labyrinth seals

with other geometry and operating conditions. They also reported that both

coefficients of direct and cross-coupled stiffness are linear functions of inlet

pressure and the direct stiffness remains negative at all operating conditions. Since

the cross coupled stiffness is a destabilizing influence, small values are desired

for this coefficient. It results that low pressure and inlet swirl ratios are preferred

from a dynamic stability point of view of the seal.

Fig. 12. Direct stiffness coefficient versus pressure

ratio with the inlet swirl ratio as a parameter.

Fig. 13. Cross coupled stiffness coefficient versus

pressure ratio with the inlet swirl ratio as a parameter.

1 2 3 4 5 6 7 8 9

-25000

-20000

-15000

-10000

-5000

Dir

ect

sti

ffn

ess c

oeff

icie

nt

K (

Nm

-1)

PR (Pin/P

out)

= 0

= 0,5

= 1

1 2 3 4 5 6 7 8 9

0

5000

10000

15000

20000

25000

Cro

ss

co

up

led

sti

ffn

ess

k (

Nm

-1)

PR (Pin/P

out)

= 0

= 0,5

= 1



Figures 14 and 15 represent respectively direct and cross coupled damping

coefficients versus the pressure ratio (Pin/Pout) with the inlet swirl ratio as a

parameter. Both magnitudes of these coefficients increase with increasing

pressure and inlet swirl ratios. The direct damping coefficient is a stabilizing

influence; hence maximum values are desired for this coefficient. Therefore, high

pressure ratio and inlet swirl velocity are beneficial from a dynamic stability

point of view of the seal.

Direct damping and cross coupled stiffness coefficients remain the two

responsible coefficients for the dynamic stability or instability of the seal. The

direct damping is a stabilizing influence while the cross stiffness is a

destabilizing influence. The obtained results in Figs. 13 and 14 for these

coefficients do not permit to be pronounced about the effect of pressure and inlet

swirl ratios on the dynamic stability of the seal. In order to better evaluate this

stability, the following whirl frequency ratio will be used.

D

k (16)

This dimensionless parameter defines, at each whirl frequency Ω, the

destabilizing influence of cross coupled stiffness to the stabilizing influence of

direct damping. A minimum whirl frequency ratio is preferred from a dynamic

stability point of view of the seal.

Fig. 14. Direct damping coefficient versus


Fig. 15. Cross damping coefficient versus


1 2 3 4 5 6 7 8 90

100

200

300

400

500

600

700

Dir

ect

dam

pin

g c

oeff

icie

nt

D (

Nsm

-1)

PR (Pin/P

out)

= 0

= 0,5

= 1

1 2 3 4 5 6 7 8 90

5

10

15

20

25

30

Cro

ss c

ou

ple

d d

am

pin

g d

(N

sm

-1)

PR (Pin/P

out)

= 0

= 0,5

= 1

2936 M. Kamouni


Figure 16 represents the effect of increasing the pressure ratio on the whirl

frequency ratio with the inlet swirl ratio as a parameter. This figure shows that

whirl frequency ratio increases with increasing pressure and inlet swirl ratios. It

can be easily seen that the whirl frequency ratio is minimal for the non-inlet swirl

case at low pressure ratios. Otherwise, this figure shows that high positive inlet

swirl velocities destabilize more the rotor of this kind of labyrinth seals for the

considered range of pressure ratio.

Fig. 16. Whirl frequency ratio versus


4. Conclusions

A model to analyse turbulent flow through short eccentric labyrinth seal has been

developed based on CFD computation. The model uses the general Reynolds

Averaged Navier-Stokes (RANS) equations along with appropriate turbulence

model. Some concluding observations from this investigation are given below.

Predictions of the pressure distribution along and around the seal are in good

agreement with measurements on the same seal.

The pressure is almost quasi equal in each cavity interior and pressure drop

mainly occurs in the left zone of each cavity at the labyrinth tooth throttling.

There is a high speed jet at the tip of each tooth and a large vortex inside

each chamber.

The recirculation zones in the seal cavities act as brakes to stop the axial

velocity of the flow through the seal and therefore to reduce the leakage flow.

The leakage flow through the seal increases with increasing pressure ratio but

it decreases very slightly with the inlet swirl ratio increasing for the envisaged

range of pressure ratio.

The leakage flow through the seal increases with increasing pressure ratio

and mean radial clearance.

The leakage flow becomes more sensitive to pressure ratio variations with

high radial clearances.

1 2 3 4 5 6 7 8 90,0

0,1

0,2

0,3

0,4

0,5

wh

irl

freq

uen

cy r

ati

o

= k

/(D

)

PR (Pin/P

out)

= 0

= 0,5

= 1



The direct stiffness coefficient is negative for the envisaged operating

conditions of this kind of seals. The magnitude of this coefficient increases

almost linearly with increasing pressure ratio and inlet swirl ratio.

The cross-coupled stiffness coefficient (destabilizing influence) and the direct

damping coefficient (stabilizing influence) increase with increasing pressure

ratio and inlet swirl ratio.

The whirl frequency ratio, as a dynamic stability indicator, increases with

increasing inlet pressure and inlet swirl velocity. The studied seal remains

more stable for the non-inlet swirl case.

High positive inlet swirl ratios destabilize more this kind of seals for the

envisaged operating conditions.

References

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http://arc.aiaa.org/doi/abs/10.2514/1.B35817

http://arc.aiaa.org/doi/abs/10.2514/1.B35817

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